Competion Problem in graph theory

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How can I prove that every graph has two vertices which are endpoints of the same number of edges? Any hints?

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Hint: You can apply the pigeonhole principle.

Let $n$ be the number of vertices. Associate to each vertex the set of vertices connected to it. There are $n$ such sets, and each of them has at most $n - 1$ elements. Thus, we can conclude that there are two sets with the same number of elements, and their corresponding vertices are endpoints of the same number of edges.